In the selection process, the first thing, obviously, would be to determine gun fit.
Something with screw-in chokes will offer great advantage since trap involves the need to use tighter chokes (example - IM), whereas Skeet (or 5-Stand) will require more open chokes (example - Skeet or IC etc).
Should the plan be to subject the O/U to high volume use then investing into something such as a used Browning, Winchester 101, SKB etc (in good condition) will serve very well.
CZ offers a few good field model shotguns and something such as the CZ Redhead or CZ Canvasback (read single trigger O/U guns, with screw-in chokes) could be acquired new for under $1k. The thing is, one would have to take into consideration an element of "risk" since those are basically field guns and may not (always) withstand the consistent pounding that is usually synonymous with clay shooting. Having said that, there are many extremely satisfied customers who can vouch for those shotguns' ability to take the punishment. The Huglu Lovers forum in ShotgunWorld dot com offer plenty of positive reviews.
Personally, I think the CZ O/U will be fine for light range duties. Check the trigger if you decide to go for one as many are often equipped with quite heavy triggers. CZ shotguns have nice wood and are well finished. The barrels are well regulated and shoot to POA.
The Baikal (also the least expensive) may be another route. Again, this is also a single trigger gun with screw-in chokes. The O/U is built robust. However, it will clearly lack the fit and finish that is visible on the CZ.
In view of the budget identfied, I would first look for a used brand name shotgun.